   Chapter 5.2, Problem 4E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# (a) Find the Riemann sum for f(x) = 1/x, 1 ≤ x ≤ 2, with four terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.) Explain what the Riemann sum represents with the aid of a sketch.(b) Repeat part (a) with midpoints as the sample points.

(a)

To determine

To find: The Riemann sum for the function f(x)=1x.

To represent: The Riemann sum for the function f(x)=1x.

Explanation

Given information:

The function is f(x)=1x

The region lies between x=1 and x=2.

Number of rectangles is n=4.

Find the width (Δx) using the relation:

Δx=ban (1)

Substitute 2 for b, 1 for a, and 4 for n in Equation (1).

Δx=214=14

Determine right end points xi using the relation:

xi=a+(i)Δx (2)

Calculate the value of x¯i for four subintervals:

Substitute 1 for a, 1 for i,and 14 for Δx in Equation (2).

x¯1=1+(1)×(14)=1+14=54

Substitute 1 for a, 2 for i,and 14 for Δx in Equation (2).

x¯2=1+(2)×(14)=1+12=32

Substitute 1 for a, 3 for i, and 14 for Δx in Equation (2).

x¯3=1+(3)×(14)=1+34=74

Substitute 0 for a, 4 for i,and 14 for Δx in Equation (2).

x¯4=1+(4)×(14)=1+1=2

Substitute x¯i for x in the function f(x)=1x.

f(x¯i)=1x¯i (3)

Calculate the value of f(x¯i) using Equation (3).

Substitute 54 for x¯1 in Equation (3).

f(x¯1)=1(54)=45

Substitute 32 for x¯2 in Equation (3).

f(x¯2)=1(32)=23

Substitute 74 for x¯3 in Equation (3)

(b)

To determine

To find: The Riemann sum for the function f(x)=1x.

To represent: The Riemann sum for the function f(x)=1x.

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